[Paper Review] Fullerene graphs of small diameter

Diego Nicodemos, Matěj Stehlı́k|arXiv (Cornell University)|Apr 7, 2016

Fullerene Chemistry and Applications4 citations

TL;DR

This paper constructs an infinite family of fullerene graphs with diameter at most $\sqrt{4n/3}$, disproving a conjecture by Andova and Škrekovski that the diameter of any fullerene graph on $n$ vertices is at least $\lfloor\sqrt{5n/3}\rfloor - 1$. The construction uses planar duals of specific triangulated graphs derived from a 6-regular triangulation, with vertices identified in a cyclic manner to form nanodisc-like structures that minimize diameter for their size.

ABSTRACT

A fullerene graph is a cubic bridgeless plane graph with only pentagonal and hexagonal faces. We exhibit an infinite family of fullerene graphs of diameter $\sqrt{4n/3}$, where $n$ is the number of vertices. This disproves a conjecture of Andova and Škrekovski [MATCH Commun. Math. Comput. Chem. 70 (2013) 205-220], who conjectured that every fullerene graph on $n$ vertices has diameter at least $\lfloor \sqrt{5n/3} floor-1$.

Motivation & Objective

• To investigate the diameter of fullerene graphs, particularly whether they can achieve smaller diameters than conjectured.
• To challenge the conjecture by Andova and Škrekovski that the diameter of any fullerene graph on $n$ vertices is at least $\lfloor\sqrt{5n/3}\rfloor - 1$.
• To construct explicit families of fullerene graphs with small diameter by leveraging geometric and graph-theoretic properties of planar triangulations.
• To demonstrate that fullerene graphs resembling a disc (nanodiscs) can achieve significantly smaller diameters than those with icosahedral symmetry.

Proposed method

• Construct the dual graph $D^*_{r,t}$ by taking two copies of a radius-$r$ ball in the infinite 6-regular triangulation, centered at poles $n$ and $s$, and identifying their outer cycles with a cyclic shift of $t$ positions.
• Define latitude $\varphi(u)$ for vertices in $D^*_{r,t}$ based on distance from the poles, with $n$ at $r$, $s$ at $-r$, and the equator at latitude 0.
• Use a closed walk $W = P_u \cup P_v$ of length $4r$ connecting $\{u_1,u_2\}$ to $\{v_1,v_1'\}$, where $P_u$ and $P_v$ are paths of length $2r$ from $n$ to $s$ through the faces of $A$ and $B$.
• Apply Lemma 2 to bound the distance in the original fullerene graph: $\operatorname{dist}_G(A,B) \leq 2\operatorname{dist}_{G^*}(u,v) + 3$, using the cut size $|\delta(P^*)| \leq 6 + 4k$ in the dual.
• Show that the dual distance between relevant faces is at most $2r - 2$, leading to a bound of $4r - 1$ or $4r$ on the original graph distance.
• Prove that the diameter of $D_{r,t}$ is at most $4r$, and since $n = 12r^2$, this yields $\sqrt{4n/3}$ as the diameter bound.

Experimental results

Research questions

• RQ1 Can fullerene graphs achieve a diameter smaller than $\lfloor\sqrt{5n/3}\rfloor - 1$ for large $n$?
• RQ2 Do fullerene graphs resembling a disc (nanodiscs) have smaller diameters than those with icosahedral symmetry?
• RQ3 Is the conjectured lower bound on diameter for fullerene graphs tight, or can it be improved?
• RQ4 What is the minimal possible diameter of a fullerene graph on $n$ vertices, and can it be achieved by a geometrically symmetric construction?

Key findings

• The paper constructs an infinite family of fullerene graphs $D_{r,t}$ with $12r^2$ vertices and diameter at most $4r$, which is $\sqrt{4n/3}$ when $n = 12r^2$.
• The diameter bound $\sqrt{4n/3}$ is strictly smaller than the conjectured $\lfloor\sqrt{5n/3}\rfloor - 1$ for all $n \geq 300$, disproving the conjecture.
• The smallest counterexample to the conjecture has 300 vertices, corresponding to $r = 5$ and $t = 1$.
• The dual graph construction ensures that all faces are triangles, and the identification of outer cycles with a shift $t$ preserves planarity and 3-connectivity.
• The automorphism group of $D_{r,t}$ is $D_6$ unless $r = 2t$, in which case it becomes $D_{6d}$, indicating symmetry variation with $t$.
• The proof relies on bounding dual distances via closed walks and applying a general distance inequality between a graph and its dual, showing that distances in the fullerene graph are at most $4r$ for $r \geq 2$.